A Surprising Pi and 5 - Numberphile

[@numberphile]

Catch a more in-depth interview with Ben on our Numberphile Podcast: kzbin.info/www/bejne/Y6Wqn5xvhMd9jc0

[@uquf]

I’m the first!

[@amritanshubarpanda1325]

When I tried various 5's in my calculator, I found out that the number of 5's you enter and start the trick, the value of 'pi' is correct. Like for example- Let t(x) = sin(1/x) So t(5) has 1 digit equal to pi i.e 3 and the rest digits are different t(55) has 2 digit equal to pi i.e 3.1 and the rest digits are different t(555) has 3 digit equal to pi i.e 3.14 and the rest digits are different t(5555) has 4 digit equal to pi i.e 3.141 and you know what..... t(55555) has 5 digit equal to pi i.e 3.1416 [ actually it should be 3.1415 but 3.14159 ~3.1416] t(555555) has 6 digit equal to pi i.e 3.14159 And so on... I really feel that this part was needed to be in this video. If you trust me, then I have no problem; But if you don't, then try out this pattern with your own calculator. Regards Yours Truly Amritanshu Barpanda

[@filip_djordjic742]

In Bosnia we do use radians in statics. Also, dots are alo used to denominate the reoccuring sequence.

[@mrnobody2873]

Calendar fractions( Degrees) were used because of astronomy since ancient times, with a degree being a day off of an alignment. Because of how integral the calendar was to agriculture, degrees were an essential way of looking at angles. Radians and Gradians are often used in math libraries for games with physics simulations, moreso space based games. It's often way eraser to calculate a metric angle internally and convert it to degrees for a UI element if necessary. The reason it it used internally is because of FPP and hard to debug rounding errors that may popup far downstream when using degrees.

[@millicentsmallpenny5837]

Its refreshing to find sanity and rationality here. Been watching some jain 108 vids. Nuff said

[@benjamintoddjohnson]

I'm a bit surprised that after learning about gradians, Brady didn't come up with a new angle measurement known as bradians.

[@digitig]

"Binary radians"? Maybe a full turn is 1?

[@Waggles1123]

@@digitig That's actually already used in measuring revolutions.

[@ShabbaDabb]

Ooooooo

[@Codricmon]

I would assume that bradians are used to measure Parker circles.

[@DerekMacaroni]

@@Codricmon No, that's a different thing. There are 3 Parker radians in a circle.

[@SoleaGalilei]

"Our choice is free, we just have to accept the consequences." Truer words.

[@neillunavat]

Who are you, so wise in the name of science?

[@themathmoth7393]

@@neillunavat ways*

[@jevicci]

Sound like Rush lyrics.

[@blueredbrick]

I love how the taylor expansion broke down with using a singe 5 :)

[@benmerkey8823]

choose 1, turns are the superior unit of angle

[@pinkdispatcher]

The military sometimes use 6400 units (called "Strich" in German, literally "stroke", or "line") to a full circle, which divides very nicely, and is also close enough to 2000 Pi to make distance estimates very simple if you have binoculars with "Strich" grading and know the actual size of objects. A thing of apparent angle of 1 Strich, that is 1 m long, is 1 km away.

[@InShortSight]

I love maths.

[@thefakepie1126]

I'm not a mathematician so I like to use 1 units (full circle , half circle , third circle , quarter circle , half quarter circle , quarter quarter circle , just a small bit , just a veeeeery small bit , ect...) 73 degrees would be : about 5 half quarter third circle minus a tiny bit I love this system , straight to the point

[@alandouglas2789]

InShortSight :3 You realise that’s its apparent size, and not equal to right?

[@hatebreeder999]

@@thefakepie1126 similar system is used in music to divide time duration of notes..whole note half note quarter note third note..etc

[@geoffroi-le-Hook]

In the US they call that a mil

[@shpensive]

There's a fine line between a numerator and a denominator

[@user-me7hx8zf9y]

thanks.

[@Aadil2Adnan]

It's called a vinculum apparently

[@javid29]

Only a fraction of people will find this joke funny

[@superscatboy]

Ha ha yes like the joke in the video ha ha

[@alveolate]

@@javid29 because this old joke is really divisive.

[@yeahuh4128]

"If there's Pi somewhere, it means that the equation is related to circles." -3B1B

[@Nah_Bohdi]

....only the Sith deal in absolutes.

[@rhythmgoel4245]

I know this one

[@dedgzus6808]

not always.

[@kurumi394]

@@dedgzus6808 Can you give us an example? Genuinely curious

[@dedgzus6808]

@@kurumi394 Euler's answer to the Basel problem is the best example I can give.

[@alttiakujarvi]

A more intuitive way would have been to stick with fractions, introduce the unit conversion and refactorize the 1/55555*Pi/180. You end up with: Pi/(55555*180) = Pi/(11111*5*180) = Pi/(11111*900). =Pi/9999900 Each added "5" in the trick adds an other "1" in the refactorization and a "9" in the final fractions. From here it is quite clear why this happens: adding "5" increases the divisor, which gets closer and closer to a full power of 10.

[@paul55604]

I'd add one more line: =PI*(1 - 1/n), where n is the number of 5s.

[@AdamSpanel]

@@paul55604 What? No. It should be Pi * (1 - (1 + ((10^n - 1) * 100)) / ((10^n - 1) * 100))

[@926prasenjit]

​@@AdamSpanel 1/180=0.0055555555... is the key here

[@926prasenjit]

@@paul55604 ​ BoltKey 1/180=0.0055555555... is the key here

[@davefoc]

@@926prasenjit Cool and it answers my question. Do more fives get you closer to pi. I think that might be obvious to somebody that can think about this better than I could, but this makes what is going on clear. Well done. Also well done to Altti Akujärvi whose comment came pretty close to this.

[@CanaanPoE]

I had a year of college trigonometry and still didn't quite understand radians, and you just explained it to me in about 30 seconds and now it just makes sense. I wish I had teachers that were this clear and concise when teaching.

[@hadz8671]

grads are sometimes used to measure latitude on maps of France, because metre was designed so that 1 grad of lat = 100km.

[@Jim73]

I can't stop wondering where that ladder goes to....

[@ericfox7021]

Up

[@tubeofglue8117]

The top

[@MasterChakra7]

To the first digit of Graham's Number.

[@PaulPaulPaulson]

Chaos is a ladder. Which means it must go straight to 2020.

[@InTheBeginningTheUniverseWas]

maybe the set of shelves is very tall

[@TheAlps36]

Thanks for explaining what GRAD means on calculators! I don't think even my maths teachers knew

[@achance75]

Gradians are often used here in Sweden in land surveying. They are sometimes refered to as new degrees but best known as gon. From the Greek word “gonia” which means angle. So trigonometry means literally three-angle-measurement. They are used mainly to simplify calculations and to avoid the need for conversions between degrees minutes and seconds which have different bases (multiples).

[@vivekg8725]

Engineers be like “I told u guys sinx =x”

[@flashpeter625]

Also tan(x) = sin(x) = x, of course.

[@sachinmysorekar5593]

Don’t forget g=10

[@Seb135-e1i]

pi = 3 = e

[@vivekg8725]

Sachin Mysorekar and no air resistances.

[@JayN_101]

vivek g 'Assume ideal gas'

[@mebamme]

4:21 I was sure Brady was about to define a new unit and call it "Bradians".

[@davidgalloway7195]

We already have Bradians. Defined in the eighties (at least) it referred to, at the time, 256 slices around a circle but later it could be 4096 or other powers of two slices.

[@aurelia8028]

With his ego, he'd _love_ that

[@Sciencedoneright]

@@davidgalloway7195 woah

[@mscha]

One thing that is missing from this explanation is in chapter two: why is the gradient of the sine function (in radians) 1, close to 0? The reason is, that the sine is basically the y coordinate from the point after walking a certain distance along the unit circle starting at (1, 0). And when you walk a tiny bit (e.g. 0.00000001π), you're basically walking straight up (to approx. (1, 0.00000001π)).

[@Gehr96]

Or mathematically more rigorous: The reason lies in the Taylor approximation of sin(x) at x=0. We have sin(0) = 0. d/dx sin(0) = cos(0) = 1. Therefore sin(x) ≈ 1 x = x. If you want a more accurate approximation you can add the next term: d²/dx² sin(0) = -sin(0) = 0, d³/dx³ sin(0) = -cos(0) = -1. Therefore sin(x) ≈ 1 x - 1 x³/3! = x - x³/6.

[@stephenbeck7222]

Gehr96 the Taylor series relies on the derivative of sine, which is what Michael is laying the groundwork for. You need the limit of sin(x)/x as x goes to 0 to be 1, which is generally proven with geometry and the squeeze theorem. You use that limit to prove the derivative of sine.

[@hepiik.8822]

9:55 In Poland we use brackets for it So it will look like 0.0(000018) and it means 0.0000018000018000018....

[@RazvanMihaeanu]

Same in Romania

[@bagratm4854]

This notation is used in all former Soviet Republics as well.

[@chwytliwanazwa4853]

myślałam że w innych krajach jest tak samo

[@hetulbhatt5787]

In India we put a bar on the recurring digits. However, while programming we use parentheses.

[@User050068]

@Adriano Andrade Всё для удобства пользователя.

[@phoenixstone4208]

shame, he clearly should be using the legendary Gaxio

[@AlKaBen]

More unboxing videos !!!

[@andie_pants]

i have no idea what that means, but have an upvote nonetheless! :-)

[@heyandy889]

Hello calculator fanciers, welcome back to another calculator review video

[@leadnitrate2194]

@@andie_pants go watch Matt Parker's calculator unboxing videos on this channel. Hilarious.

[@AlKaBen]

@@andie_pants it's an inside joke, there is a hilarious video with matt parker unboxing calculators.

[@bemusedindian8571]

I have said this before, Ben Sparks has the best Numberphile videos. Period.

[@Giantcrabz]

neil sloane is pretty great

[@BaronSamedi1959]

I like "mills" to measure angles. 1 mill is the angle of 1 m seen at a distance of 1km. It makes it easy to convert angles into lengths at a certain distance (multiply the angle in mills by the distance in km and you get the length of that angle at that distance). Why would you do ever want to do that? Artillery observers use it to correct the fall of shot. You know the target is 1.5 km away and the first short fell 40 mills to the left of the target. You correct the next shot with a "Right 60"(40 x 1.5) and you should be fine. Of course, that doesn't take into account the errors in each shot, so you actually don't correct until after a number of ranging shots. Then you add (or subtract) all errors and make one average correction. Also, to get the distance of the shot right, you use a binary search algorithm: your first correction is always a 400m "jump", then 200m, then 100m and finally 50 for the "fire for effect". And finally, you compare the fall of each shot with some kind of "standard deviation" (called "F a" or "Fourchette Apparente"). If the shot falls outside of the "F a" measurement, then you don't take it into account as it was not part of the same "statistical family" of shots and shouldn't be used. Being a forward observer, really made you use trigonometrics, mathematics and statistics in a practical way. And no calculators used! You had about 5 seconds to do all these calculations in your head and give the order to correct the next shot. You should make a numberphile about such practical uses of mathematics. (To give some background: that was how it was done up to the 1980s. Now it is all laser range finders and computers, bit I was trained to only use a map, a magnetic compass and binoculars with crosshairs graduated in mills.)

[@Ultiminati]

that's cool

[@KnaveRain]

Man that is amazing, I never knew so much work went into that. I wish they would integrate things like these into the curriculum. Im not a huge fan of math, but thats a super cool application.

[@sluggermendoza9903]

Whoa, finally an explanation for the stuff they do with arty in war movies. Thank you! I had some idea of the corrections using mills, but I had no idea you had to think about the standard dev. and averaging all the corrections for the fire for effect.

[@anotheraggieburneraccount]

Sounds like .001 radians

[@PC_Simo]

@@KnaveRain I’m pretty sure they would do stuff like that, in the 3rd Reich: Teach students the derivatives of Maths, for military purposes. They literally had a class for making paper planes.

[@whydontiknowthat]

There’s a nice generalization of this to numbers other than 5. It turns out that if you use the number n (where n is a positive integer), then the decimal expansion tends to 5pi/n.

[@bazoo513]

This was fun, and Ben obviously enjoyed leading us through it. Well done, guys!

[@orsettomorbido]

"It's Pi enough to cause a reaction in anybody who has seen Pi before" HAHAHA

[@nekogod]

I would have thought one of the strongest reasons for using 360 is the number of factors it has, 1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,30,36,40,45,60,72,90,120,180,360 which makes dividing up circles much easier.

[@agustingomez7142]

Every video with Ben Sparks blows my mind!

[@alansmithee419]

So Ben Sparks your interest does he?

[@nithinanand2517]

Growing up I thought the gradient was just like he defined, circle chopped into 400 parts or a right triangle chopped into 100. But I think the better way to look at it is by the meaning of gradient, as in slope. So when we put G mode in the calculator, we are essentially getting the slope of line in percentage. 10% means, for every 10 steps you take, you go up by 1.

[@ximecreature]

Oh wow surprised to see the word vinculum here. Last time i've seen it was at the university, studying law. Vinculum iuris was the legal bound between parts of a contract in ancient Rome [Gaius]. Vinculum is a bind. The numbers here are "bound" together to be repeated. That one will be easy to remember for me !

[@laurak1545]

These mathematicians all seem so freaking happy

[@WillToWinvlog]

it's cuz they are being paid to share recreational math

[@chrisengland5523]

Only when they've found the answer.

[@DavidMcCoul]

And you're not?? When numbers work out beautifully it's a wonderful thing!

[@oj.b.3889]

I called this the perimeter constant Ω = sin(π/X)*X Where X is no. of sides Therefore perimeter is 2ΩR Where R is length of line between center and vertice Hence lim X→∞ sin(π/X)*X Approaches π

[@MrForreststarr]

I don’t know if I’m more satisfied with doing it myself,... or the fact he brought out the brown paper screen to explain it right at 3:14.

[@cerwe8861]

If you see π, a circle isn't far.

[@N.I.R.A.T.I.A.S.]

I guess you just have to wait until they get around to it, ba doom tssh.

[@XerosXIII]

You can almost smell it.

[@joonatan003]

@@XerosXIII 🤣

[@ajc1476]

@@theprofessionalfence-sitter and from where do radians and degrees come from?

[@XenoghostTV]

@@theprofessionalfence-sitter Bruh

[@LaGuerre19]

Always liked Ben Sparks videos, but now he is, to me, a member of the Numberphile Pantheon which includes Cliff Stoll, James Grime, and Holly Krieger, etc. What an excellent maths communicator!

[@Veptis]

In school we did a class trip to do practical trigonometry. Essentially surveying the land around. So we used theodolites. Half of which were in Grad (360° degrees with minutes and seconds) and the other half in Gon or Neugrad(I guess gradians since it was 400 with decimal places). And you had to be careful to use the correct transformations for the final calculations of where to plot the features

[@Qtini]

In that case we should expect tau to appear with repeated 25s. Or I suppose 2 followed by 7s and ending in a 5 due to the 2 digit carryover.

[@kc_ee]

This was the best recent video that you guys have put out. Props.

[@MrDowntownjbrown]

Could you do a video on the items in the background? Kind of a Numberphile/Objectivity crossover. Thanks for all the videos during this time!

[@davidborger7159]

As far as I know, ancient mathematicians used 360 as it had a lot of factors((2^3)*(3^2)*5)making eventual divisions easier. Also, babilonians used a base-60 numerical system, so they might have used 360 as well

[@Ddub1083]

Yes. It is known as a highly composite number. Any number that has more factors than any number lower than it. They are also called "anti-primes"

[@RazvanMihaeanu]

The answer is Pythagorean triple: 3x4x5=60 Lot of space to turn around in that confined space...

[@hamiltonianpathondodecahed5236]

btw another possible explanation is that the earth takes about 360 days to revolve around the sun hence the people in the ancient times would have taken it for convenience

[@N.I.R.A.T.I.A.S.]

David Borger Interesting to know that that was a factor in their decision making, ba doom tssh.

[@anticorncob6]

It's also a "superior highly composite number", which are a special subset of the highly composite numbers. I wish numberphile made more videos about highly divisible numbers, kind of like how they make lots of videos about really huge numbers. It's an interesting topic.

[@yuryschkatula9026]

Fantastic! The more times the magic "5" is repeated in the denominator, the more zeroes then appear between "18" and next "18" at the fractional part. What a Pi'etic fact!

[@acetate909]

@2:02 "Yaaaa, pie like"

[@Dr.JudeAEMasonMD]

This is bowing my mind right now! 4:45

[@567secret]

Pretty sure I worked it out, spoilers below: Basically it's to do with the small angle approximations and the conversion between radians and degrees, since pi/180 is the radians per degree and you're using a number such as 1/5 or 1/55 or 1/555 etc. you end up with something like pi/99900 radians. Since small angle approximations state that for radians sin(dx) ~ dx then you end up with a result of about pi/99900, since we use base 10 this is roughly equal to pi*10^-5. Edit: I showed it a similar way, but I think my way is clearer.

[@EebstertheGreat]

The computation is sin(1/555555°) = sin(9/(5 × (10⁶ - 1))°) = sin(π/(100 × (10⁶ - 1)) rad) = sin(π/(10⁸ - 100) rad) ≈ sin(π × 10⁻⁸ rad) ≈ π × 10⁻⁸.

[@926prasenjit]

1/180=0.0055555555... is the key here

[@lynk_1240]

I have to say, even having degrees in both physics and engineering, I have never before encountered before today what the 'grad' on my calculator was. Thank you for enlightening me.

[@SigmaThoughts]

Unexpected pie is my favorite kind of pie

[@jackys_handle]

Gradient is commonly used to assert the steepness of roads as a percentage. 50% means a 45°

[@therizinosauruscheloniform9709]

sin(x) = x * laughs in engineering *

[@doomdoot6731]

Only for smol angles though. Also iirc the step where you assume that to simplify differential equations is called "linearisation" or something along those lines. Because you turn a sine function into a linear function.

[@panonf.9106]

@@doomdoot6731 cos(x) = 1

[@integralboi2900]

Sin x = tan x = x

[@Ohmau33]

*physicists have entered the chat*

[@Jet-Pack]

@@Ohmau33 pi=3.14

[@nate-wilkins]

I thought I knew what radians were and wow the explanation was way better than how I learned them.

[@monlewi1976]

7:08, damn, the most accurate handwritten sin wave

[@pranavlimaye]

Not really, sine waves are a lot flatter than that. Remember, 2π=6.28 which is thrice as wide as y going from +1 to -1

[@monlewi1976]

We don't see notation on axis, so Y could be more stratched compare to X, and yet graph perfectly align with sinwave

[@hamgelato8143]

ah, it's happy tim

[@numberphile]

Kudos for being a Numberphile Podcast listener too!

[@seeseefok7659]

:D

[@anneling529]

Ha! I really enjoyed that podcast.

[@doodlegoat]

555555 = 5/9 × [10^6 - 1]. If you do your evaluation with that construction, you don't need your repeating decimal with 18 in it and all that hand-waviness at the end.

[@rocketpig1914]

Guess they decided that was a little too much detail to take in given all the rest.

[@e.b.1115]

Yes, exactly what I did!

[@therealax6]

They way I did it in my head after pausing was to "round" it to 555555.55555..., since it's an approximation after all (and this is the way it is built as you add more fives). That makes the -1 go away (that's just 5/9 * 10^6) and everything cancels out nicely.

[@gergelykiss]

9:31 In Hungary we also use dots for denoting recurrence.

[@JNCressey]

2:28 "we haven't mentioned a circle yet" but... "sin"

[@RFC3514]

By that logic, you're "mentioning a circle" whenever you measure the angle between two straight lines. The fact is, he _hadn't_ mentioned a circle. He (and most viewers) obviously knew that trigonometric functions (despite being derived from triangles) are closely linked to circles.

[@black_platypus]

"repent"

[@ravindrawiguna8681]

@@RFC3514 kinda... ish

[@PC_Simo]

@JNCressey My thoughts, exactly 🎯!

[@SimBol1216]

Interesting to see Russell Crowe teaching mathematics.

[@neelparmar6690]

My first thought!

[@concretemathematics8146]

I thought that was Seth Rogen teaching Math.

[@evertvandenberghe]

Roflmao

[@AvalonWizard]

Would have been better if he had Russell’s accent.

[@colenelson658]

He really has a Beautiful Mind.

[@samueldevulder]

Grad is pretty useful for navigation. Perimeter of earth is 40 000 km. So 1Grad is 1/400 of the 40 000km.=100km. Perfect metric division. If your boat moves 100km on the equator, celestrial objects moves 1grad in the sky. If you measure that an object moved 0.1 grad beween two measures (after correction of earth rotation), your boat has just sailed for 10km. Fairly cool, isn't it ?$

[@KarenSDR]

This video illustrates why I think math and humor can be alike. When someone tells you the kind of joke where you pause, and then it hits you and you start laughing, it's because there's a kind of delight in suddenly understanding an unexpected connection. That happens about halfway through this video. The first part of the "joke" is when he shows a surprising result on a calculator. But the part that won a delighted grin from me was about halfway through when I started to see why it works.

[@GabrielPohl]

Engineers be like: π and 5? I don't see any difference

[@raphaelkelly861]

Ceil(e) = floor(π) = 3

[@ethandavenport4310]

Cosmologists be like: π and 10? I don't see any difference

[@TruthNerds]

@@raphaelkelly861 π=3 for sufficiently small values of π and sufficiently large values of 3.

[@RWBHere]

Funny that you say that. We had an electrotechnology lecturer on our degree course who routinely rounded complex equations by using the approximations (Pi) squared = 10, and g = 10 N/sec squared, then cancelling them all out with any tens on the other side of the equation. Of course 10 = 2 x 5. This can turn some very complex equations (to a mathematician) into simple mental arithmetic for an engineer. His results were never wrong by more than 5%, and generally closer than 1%, which is usually more accurate than the results which most real-world situations can be expected to give. Most people using a rule, for example, never read hundredths of a millimetre, and only really finicky chefs measure ingredients to much closer than 5%. In wafer fabrication (my field), you can calculate all parameters to the finest possible, but the resulting semiconductors will have a normal distribution of characteristics which can range from near-perfect to unusable, across a wafer, simply because of semiconductor imperfections, temperature gradients, thermostat variations, the accuracy of etching and doping chemical molarity, operator techniques, degree of cleanliness, etc. In reality, two identical robot systems, or two people, working together, following exactly the same procedures, in time with one-another, and using the same opting ovens simultaneously will have different results, which only rarely match within 1%.

[@TruthNerds]

​@@RWBHere I don't think OP doubts that engineer calculations are accurate enough to give usable results. It's still funny to think about it from a mathematical perspective, though, where you are expected to be 100% accurate, e.g. you would get a point deduction in a math test iif you substituted 355/113 for π, even though the relative error is less than one in ten million.

[@jacksparrow440]

5:37 what's to notice though that, as opposed to r, theta is *dimensionless*. For example: r has the dimension of a length, pi*r² has a dimension of an area (i.e the square of a length), 4/3*pi*r³ has a dimesnion of a volume (i.e the cube of a lenght); but 2*pi*r has the same dimension as r (i.e a length), which means the measure of an angle has no dimension. This is not to be confused with units: units of measure are abritrary and useful on a daily basis, whereas the dimensionality of a measure is more fundamental

[@scottriley5141]

That's why there are no SI units for angles (some list them as 1). Of course, m/m cancels.

[@litchqueenasenath5995]

"Pi/4 radians of pizza please." "What is that, like a slice?!" "More like exactly a slice!"

[@davidwuhrer6704]

I prefer π·𝑧·𝑧·𝑎 units of pizza, where 𝑧 is the radius, and 𝑎 is the height.

[@RWBHere]

Six radians, please. And you can keep the change!

[@abangfarhan1]

10:08 "they say there's a fine line between a numerator and the denominator" lol

[@samiraperi467]

I like unexpected pie.

[@PC_Simo]

”How many times would you like me to type the digit: ”5”?” *Me, a Cabtist:* ”3.”

[@givrally]

I haven't watched past the 2 minutes mark, but here's my attempt at an explanation. Spoiler alert, probably ? Since sin(x) ~ x for small values, I don't think it's in radians, otherwise you'd get something close to 1.8, which, to the best of my knowledge, isn't π. Another thing is that 1/(55555...555) is approaching 1/0.555555... multiplied by 10 to some power. 0.555 = 5/9, so the reciprocal will approach 9/5 (= 1.8) multiplied by 10 to some power, or 180 multiplied by 10 to some power minus 2. If the calculator is not in radians, it's probably in degrees, so you have to multiply by π/180 to get it right, which cancels to get sin(π times 10 to some power), and since it's getting smaller and smaller, the final answer is getting closer and closer to π times some power of 10. Am I right ? I don't know, I'm going to watch the rest of the video now. EDIT : Damn right.

[@nicholashylton6857]

In grade school, radians made more sense to me. In tests, those were the first questions I solved, but struggled with the rest.

[@reox42]

Theodolites sometimes use gradians, I recently saw one with a scale in gradians. Looks like the gradians are also useful for stepper motors, as they have for example 200 steps per rotation.

[@MeltedMask]

Gradians make mental math easier in practical applications and that is reason why its used by land surveyors in europe. Addition and subtraction of numbers: 45, 90, 180, 270 in degrees Vs. 50, 100, 200, 300 in gradians.

[@danielbarreiro8228]

As with most units in the SI or metric system, gradians are not so arbitrary. Sailors know that a nautical mile is about a minute of arc over a meridian. Given the circumference of the Earth, 40000km / 360 / 60 = 1,852km, which is a nautical mile in kilometers. Likewise, a 'minute' of a grad gives you a kilometer, assuming 400 grads to a circle and 100 'minutes' to a grad: 40000km / 400 / 100 = 1km. I put quote signs on the 'minute' because grads just have decimal fractions, there are no minutes (or seconds for that matter) but centigrads which, BTW, is the reason why giving temperatures in degrees Celsius is preferred to degrees centigrades.

[@Gamedolf]

I first stumbled across this when I was playing with my calculator in year 8/9 and wondered why x*sin(180/x) approached pi as I put bigger numbers in for x, didn't get the answer from my teacher at the time but this explains it

[@redpepper74]

tbird81 In middle school all three times that I asked my history teacher what the difference between Republicans and Democrats were, he just said “there’s a lot of differences.” Is modern history just too scary for me to handle? Like I pressed him but he didn’t want to give me the simplest overview.

[@JayN_101]

The game is all about knowing your *limits*

[@scottdobson1276]

Gradians are used in road design. You see signs on steep down hill sections that say x% grade.

[@johnfenske7764]

2:25 "We haven't mentioned a circle so far" but isn't referring to sine inherently referring to circles?

[@Lightn0x]

It is, yes. That's why I didn't find this fact to be as "surprising" in the first place. You're working with trigonometric functions, and pi pops up, that's more or less expected.

[@kylebryancagasan4447]

Correct me if I'm wrong but aren't the trigonometric functions derived from triangles? You can mention trigonometric functions in the absence of circles.

[@miguelangelmartinezcasado8935]

@@kylebryancagasan4447 yeah but tell me about a triangle that can't be build by circles. It's kind of a cheat to use the sine

[@martinepstein9826]

"Isn't referring to sine inherently referring to circles?" I wouldn't say so. When you refer to squaring a number are you inherently referring to regular 4 sided polygons?

[@RodelIturalde]

@@kylebryancagasan4447 trigonometry is mostly about a unit circle. And derivation if most formulas comes from clever usage of said circle.

[@micaelaroyo4837]

that was seriously so cool! They should teach these sort of cool trick to kids in high school because many thing that math is boring but if they we're to understand the overall trigonometry, then this would blow their minds

[@GregB314]

Fascinating stuff. And Ben is excellent at explaining.

[@Kenchow96]

Fun fact: 1/5 = 1 * 1/5 1/55 = 1/11 * 1/5 = 0.0909... * 1/5 1/555 = 1/111 * 1/5 = 0.009009... * 1/5 So you're getting ever closer to 0.9 * 0.2 * 10^n Or 180 * 10^m With the significance of 180 explained in the video (DegRad) Also why the 1/5 case fails!

[@anneling529]

Great video! I always teach my 6th grade Pre-Algebra students the words "vinculum" (and its plural, "vincula") and "repetend" and tell them they can impress people by using them at the next cocktail party they attend. (I've taught Latin, too, so I explain the literal meanings whenever I can, which is actually quite frequently in math!) Thanks to this video and the comments I've just read, I can now explain the alternate representations with dots above the beginning and end of the repetend and using parentheses around the repetend. I also never knew that a fraction bar is also called a vinculum in some parts of the world. Very interesting! I think I'll start teaching my wee ones about radians and gradians, too. Or maybe I'll just show them this video and let Ben do it for me... Also, I agree with other viewers that a Bradian (Bradyan?) should be a unit of measure and that Happy Tim needs to make a series of videos about all those cool nerdy things on the shelf behind him!

[@chrisengland5523]

If you start talking about Latin or maths at a cocktail party, you'll probably find you're talking only to yourself for the rest of the evening.

[@battledraw]

Those sound effects make everything way cooler

[@camilohiche4475]

"Our choice is free, we just have to accept its concequences."

[@mydemon]

"Yeahhhhhh........ Pie like...." lol

[@adamgreene9938]

10:18 That long calculator makes me uncomfortable...

[@SquirrelASMR]

Lots of these tricks I've seen where pi comes out magically without circles, but still has had a sine or cosine function, like that bouncing pool ball question giving pi-like colisions

[@marksusskind1260]

It says 55 comments on this page when I loaded it, but I loaded it half an hour or so ago. 5/9 is zero point repeating-5. 9/5 is a term I use for converting between Fahrenheit and Celsius, so I see 1.8 a lot.

[@krystofdayne]

Yeah for that Fahrenheit conversion, I use a much more hand over fist calculation. If you want to convert Fahrenheit to Celsius correctly, you would first have to subtract 32 and then multiply by 5/9, so 100°F=[(100-32)*5/9]°C=37.7777....°C. Which is a fairly horrible calculation to do if you just want a quick approximate conversion. So what I do is just subtract 30 and divide by half, so in that case 100°F=[(100-30)/2]°C=35°C. That's close enough to approximately know what sort of temperature range we're talking about. And when the temperature gets really high, you can start ignoring the subtraction and just approximately take half to get Celsius. But mostly I use that -30, divide by 2 calculation. I guess that could work the other way round too, so to get from Celsius to Fahrenheit, multiply by 2, then add 30.

[@he1986]

I also use 1.8 when converting between knots and m/s. Not so strange then, as knots is tied to the earth’s circumference, or 360 degrees...

[@MK73DS]

Because of the repeating pattern, the error between pi x 10^-k and sin(1/555..55) (with the right k so that the error is minimal), is also very close to pi x 10^-l for some l > k.

[@panonf.9106]

Look it's Tim, but smiling!

[@ro-ce8vg]

I’ve watched a fair amount of these videos but this was the first one I actually tried figuring it out before watching the explanation and got it

[@EnigmacTheFirst]

A surprising “15 seconds ago” video

[@vynneve]

NOTE: This is my personal explanation before I saw the video's. Thought it could be cool to share it. Almost the exact same, except for the final stretch to prove it's π*(negative powers of 10) This is just...an expected π to me lol. It wasn't working for me, then I was like oh we are taking it as degrees. But calculators don't actually use degrees...they convert to radians regardless, it's just easier for you. And with a small x: sin(x) ~ x And if you remember how to convert angles, it's very obvious immediately. sin(1/5555... * π/180) ~ 1/((180)(555...)) * π

[@Fosgen]

If you knew the magnificence of the three, six and nine, you would have a key to the Universe.

[@l3alamiya]

Gradian is used in land surveying

[@SteveBakerIsHere]

In the early days of 3D computer graphics we use what we called "brads" (binary radians I guess) which ran from 0 to 256 - handy because an angle fits into a byte and you don't need such big lookup tables for trig calculations - also when you get large angles that go beyond a full circle, you can just chop off the high order bits and the angle is always between 0 and 255.

[@trevcam6892]

Mathematics was my best subject in school followed by 50 years of using mathematics as Engineer and I'm still learning something new. Thanks!

[@chrisengland5523]

Same here, but the thing that I now realise is that what I'm learning now is the same as I learnt 50 years ago but in the meantime forgot.

[@BigMcLargeHuge19]

ill never forget my calculus teacher writing “sin x ≈ x” and asking us to show that it is true in this specific case

[@davidgillies620]

sin _x_ ~ _x_ for _x_ small is a very useful thing to know. Also, any time you see repdigits, think (10^n - 1)/9 (because 9/9 = 1, 99/9 = 11 etc.). So the exact value for _n_ fives is sin(pi/(100(10^ _n_ - 1))) which goes to sin(pi/(10^( _n_ + 2))) for _n_ large..

[@douglasbrinkman5937]

20% off - that's 1/5th off!

[@Aadil2Adnan]

Now that's brilliant

[@davemarm]

80% on - that's 4/5 on!

[@AlanTheBeast100]

Metricish: mils: 1/6400 of a circle. (artillery unit: missed a target @ 1000m by 30 m to the left? Correct the angle by 30 mils to the right). [6400 replaces 2pi*1000.]. {Swedes used 6300 which is better, but deprecated}

[@NabeelFarooqui]

My calculator doesn't have the repeating dots :(

[@AuroraNora3]

@@diptoneelde836 It's also a calculator-specific thing

[@redpepper74]

Hoo Dini So I can’t bring my Texas Institute calculator to Britain and watch the vinculum turn to a couple of dots? Sad 😞

[@PopeLando]

I'm British, and I never heard of the two dots thing before now. I've only seen the bar version.

[@pierrotinturquoise]

I'm Bangladeshi and I have never heard of the bar thing before now. I am seeing that dots since seventh grade. I think the whole Indian subcontinent uses dots instead of bars. From tea to dots, sometimes I think we are more British than brits.

[@slolilols]

Indians actually use both dots and bars, just the bars are more frequently used. :)

[@immortalsofar5314]

"Degrees are an arbitrary number and you don't have to use it" Damned right! Back in the 8 bit days, I divided circles into 256 parts - simple lookup (sine_table+idx), auto-repetition (255+1 = 0) and it fits into a single CPU operation. I also used fixed point arithmetic so degrees x (sine x 256) gives the 8 bit result in the high byte. Plus, being a power of 2, 1/2, 1/4, 1/8 are very easy to calculate. After I got used to that, going to degrees seemed ridiculously arbitrary and radians were fine so long as it stayed in Pi units and didn't have to be converted to a numeric result. Then again, Pi = 128 meant that converting to radians was ridiculously easy, again using fixed point arithmetic.

[@Maniclout]

The vinculum, never heard of that one!

[@Sargentleman]

To discover the 5s, it's slightly easier: pi = sin(pi * (1/x)/180) ~= pi * (1/x) / 180 pi = pi * (1/x) / 180 x = 1/180 = 0.0055... (repeating 5) Multiply the approximate equation by 10^-n and change variables pi * 10^-n = pi * 10^-n * (1/x) / 180 pi * 10^-n = pi * (1/(x * 10^n)) / 180 pi * 10^-n = pi * (1/y) / 180 For n > 3: y = 0.0055... (repeating 5) * 10^n y = [(n-2) digits of 5].55... (repeating 5) Approximate with the floor. y ~= [(n-2) digits of 5]

[@alicehancock1]

Do I see a Klein bottle behind him? *Cliff stoll intensifies*

[@j.503]

He presented a really intriguing puzzle and then explained it so that it became completely demystified.

[@tracyh5751]

2:22 "But we haven't mentioned a circle anywhere..." You literally plugged a number into one of the circle functions.

[@redpepper74]

Tracy H I love it, I’m going to call them “the circle functions” from now on

[@Kycilak]

Exactly my thoughts. Great minds think alike, I guess.

[@ZipplyZane]

Sine is more commonly thought of as a triangle function. There's a reason it's called trigonometry, not circlometry. (A trigon is another word for a triangle. It's like pentagon.)

[@tracyh5751]

@@redpepper74 Thanks, but I'm not the first to call them that. Circles and trigonometry are intimately related.

[@Giantcrabz]

​@@tracyh5751your mother and I are intimately related

[@opl500]

More like 1/5555555 radians is pi/10^x, the sine is just a passthru function at those values.

[@Penrose707]

This professor absolutely reminds me of Russel Crowe.

[@djmips]

Don't forget BRADians or BRADs which is 0 -> 2^n-1 (degrees). On an 8 bit computer it would be 0-255 degrees and has the nice property that using AND 0xFF acts like the modulo since degrees wrap.

[@rosiefay7283]

"Gradians". Brady let slip the opportunity to suggest Bradyans.